Dark Souls 2 Soul Memory Calculator / Visualiser

In general, you can't connect between NG and NG+ at all. However, NG+, NG++, and so on can connect to each other. Since the SotFS patch, there are no restrictions based on playthrough.

You can only see White Sign Soapstone summon signs when human and there is still a boss alive in the area you are in. Small White Sign Soapstone summon signs will show up even after the boss is dead.

Being in the Heirs of the Sun Covenant doesn't increase the SM range, it simply makes your phantom/shade and sign appear golden. Whether it increases the chance of your sign appearing is unconfirmed.

The Name-Engraved Ring only seems to expand the SM range for co-op summons, but the filtering effect works with the Red Sign Soapstone and Dragon Eye too.

You can't summon or invade in Grave of Saints and Doors of Pharros, these areas are exclusively for Rat King Covenant PvP.

You can only summon one phantom or shade in Belfry Luna and Belfry Sol, presumably to prevent you ganking with 3v2 fights.

The Blue Sentinels arena has no restriction on Soul Memory.

Whether the Brotherhood of Blood arena has restrictions is currently unknown.

Abyss invasions are unconfirmed, but you can assume they work the same as regular Cracked Red Eye Orbs for now.

Mirror Squire invasions (coming out of the Looking Glass Knight's shield in NG+ and above) seem to ignore the NG/NG+ divide, although this is unconfirmed. Otherwise you can assume they work the same as the regular Red Sign Soapstone.